Okay, to give a explanation about Martin and some more:
Alice found Martin and asked him about the rest of his household. He told Alice that his tennant is probably with his "Whore of a landlady" and to fuck off as well because he sure as hell never wanted a female tennant. Alice found out where Becca lives because this was her only lead. She found her address and decided to just go for it. At this point of the story the landlord still wasn't sure where MC is, he just assumed. When he found out that his (ex) girlfriend cheated on him and he moved away he decided to ask around if anyone knew where he was. He tried this in the hope to see him again after he was diagnosed with his liver problems. Knowing that he beat up the new boyfriend of his ex, Elaine made MC call him to make sure he stops looking for him. This is when we find out that he is dying, something that even Elaine didn't know.
A bit further back in the past of MC when he was a child his father almost killed one of his co-workers. That's when the police showed up and Becca heard everything. That was the moment Martin threatened to kill her if she doesn't fuck off. Elaine found out how far his reach can go a few years later, deciding to not intervene and hope that MC can at least grow up without being physically abused. Once MC turned 18 and she found out that he moved out she didn't think about him too much anymore, Elaine thought that he finally moved on and that he probably hates his landlady and this part of the household anyway. A few years later the girlfriend cheated on him, he lost his job and got the eviction notice. Elaine knew that he didn't have anybody and she sure as hell didn't want him to move back with his landlord. She was able to get someone to slip an old picture of MC and Becca into one of his books, that's when the returning back to Becca thought crossed his mind. As soon as Elaine found out that he is on his way to his landlady she stopped everything regarding investigations for him. But then Alice showed up and well... I explained in the beginning what happened there. Elaine thought that his landlord surely will look for him when Alice doesn't return, she had no idea he would look for him for different reason, just that he eventually will do so.
I hope that explained it a bit
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